The set of all values of k for which `(tan^(-1)x)^3 + (cot^(-1)x)^3 = kpi^3,x in R,` is the interval :

Options

A. `[1/32,7/8)`
B. `(1/24,13/16)`
C. `[1/48,13/16]`
D. `[1/32,9/8)`

Answer

To solve the problem, we start with the equation:

\[ (\tan^{-1} x)^3 + (\cot^{-1} x)^3 = k\pi^3 \]

where \( x \in \mathbb{R} \).

### Step 1: Use the identity for the sum of cubes
We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Let \( a = \tan^{-1} x \) and \( b = \cot^{-1} x \). We know that:
\[ \tan^{-1} x + \cot^{-1} x = \frac{\pi}{2} \]
Thus, we can rewrite the equation as:
\[ (\tan^{-1} x)^3 + (\cot^{-1} x)^3 = \left(\frac{\pi}{2}\right)\left((\tan^{-1} x)^2 - \tan^{-1} x \cdot \cot^{-1} x + (\cot^{-1} x)^2\right) \]

### Step 2: Substitute and simplify
Substituting \( a + b = \frac{\pi}{2} \) into the identity:
\[ (\tan^{-1} x)^3 + (\cot^{-1} x)^3 = \left(\frac{\pi}{2}\right)\left((\tan^{-1} x)^2 + (\cot^{-1} x)^2 - \tan^{-1} x \cdot \cot^{-1} x\right) \]

### Step 3: Find \( (\tan^{-1} x)^2 + (\cot^{-1} x)^2 \)
Using the identity:
\[ (\tan^{-1} x)^2 + (\cot^{-1} x)^2 = \left(\frac{\pi}{2}\right)^2 - 2\tan^{-1} x \cdot \cot^{-1} x \]
We can express \( \tan^{-1} x \cdot \cot^{-1} x \) in terms of \( t = \tan^{-1} x \):
\[ = t \cdot \left(\frac{\pi}{2} - t\right) \]

### Step 4: Rewrite the expression
Now we can express the original equation in terms of \( t \):
\[ (\tan^{-1} x)^3 + (\cot^{-1} x)^3 = \frac{\pi}{2}\left(\frac{\pi^2}{4} - 2t\left(\frac{\pi}{2} - t\right)\right) \]

### Step 5: Find the range of \( k \)
The expression simplifies to:
\[ \frac{\pi^3}{8} - 3t\left(\frac{\pi}{2} - t\right) \]
To find the minimum and maximum values of this expression, we can differentiate with respect to \( t \) and find critical points.

### Step 6: Evaluate critical points
The critical points occur when \( t = \frac{\pi}{4} \). Evaluating the expression at \( t = \frac{\pi}{4} \):
\[ \text{Minimum value} = \frac{\pi^3}{32} \]
And for the maximum value when \( t \) approaches \( 0 \) or \( \frac{\pi}{2} \):
\[ \text{Maximum value} = \frac{7\pi^3}{8} \]

### Step 7: Set the range for \( k \)
Thus, we have:
\[ \frac{\pi^3}{32} \leq k\pi^3 \leq \frac{7\pi^3}{8} \]
Dividing through by \( \pi^3 \):
\[ \frac{1}{32} \leq k \leq \frac{7}{8} \]

### Final Answer
The set of all values of \( k \) is:
\[ \left[\frac{1}{32}, \frac{7}{8}\right] \]